%I #13 May 15 2021 15:52:27
%S 1,0,0,1,0,0,0,0,1,0,1,0,0,0,1,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,0,0,0,
%T 1,0,1,0,0,0,1,0,0,0,1,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,
%U 1,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,1,0,0,0,1,0,0,0,1,0,0,0
%N Triangle read by rows: T(n, k) = 1 if ((k mod 4 = 2) and (n mod 2 = 1)) or ((k mod 4 = 0) and (n mod 2 = 0)) else T(n, k) = 0.
%C The T(n, k) triangle is related to the Kn1p sums of the ‘Races with Ties’ triangle A035317. See A230447 for the Kn1p sums and see A180662 for the definitions of these triangle sums.
%C The row sums lead to three sequences and they can, quite surprisingly, be linked with Alcuin’s sequence A005044, see the formulas.
%H E. Mendelson, <a href="http://www.jstor.org/stable/2690085">Races with Ties</a>, Math. Mag. 55 (1982), 170-175.
%F T(n, k) = 1 if ((k mod 4 = 2) and (n mod 2 = 1)) or ((k mod 4 = 0) and (n mod 2 = 0)) else T(n, k) = 0.
%F sum(T(n, k), k=0..n) = A008624(n) = A026922(n+2) = A059169(n+3) = A005044(n+3) - A005044(n-3).
%e The first few rows of triangle T(n, k), n >= 0 and 0 <= k <= n.
%e n/k 0 1 2 3 4 5 6 7
%e ------------------------------------------------
%e 0| 1
%e 1| 0, 0
%e 2| 1, 0, 0
%e 3| 0, 0, 1, 0
%e 4| 1, 0, 0, 0, 1
%e 5| 0, 0, 1, 0, 0, 0
%e 6| 1, 0, 0, 0, 1, 0, 0
%e 7| 0, 0, 1, 0, 0, 0, 1, 0
%p T := proc(n, k): if ((k mod 4 = 2) and (n mod 2 = 1)) or ((k mod 4 = 0) and (n mod 2 = 0)) then return(1) else return(0) fi: end: seq(seq(T(n, k), k=0..n), n=0..13);
%t Flatten[Table[Which[Mod[k,4]==2&&Mod[n,2]==1,1,Mod[k,4]==Mod[ n,2]== 0,1,True,0],{n,0,20},{k,0,n}]] (* _Harvey P. Dale_, May 15 2021 *)
%K nonn,easy,tabl
%O 0
%A _Johannes W. Meijer_, Oct 12 2013